The conversion of light from one frequency to another in optical crystals using induced quadratic nonlinearities proceeds by a three-wave interaction process which has been described by Bierlein and Gier in U.S. Pat. No. 3,949,323.
Briefly, the conversion of light from one frequency to another using the quadratic nonlinearity term of the induced polarization in a medium (such as a crystal) having nonlinear optical properties gives rise to the phenomena of sum and difference frequency mixing. This is a catalytic process as it occurs with essentially no exchange of energy between the medium and the electromagnetic fields. Energy convervation requires that the three frequencies involved, herein called the pump (.omega..sub.p), signal (.omega..sub.s) and idler (.omega..sub.i) frequencies, meet the condition: EQU .omega..sub.p =.omega..sub.s +.omega..sub.i
For efficient energy conversion to occur, it is also required that the momentum or phase velocity of the interacting electromagnetic fields be matched while propagating through the medium. This phase matching requirement is defined by the k vectors of the individual waves as: EQU .DELTA.k=k.sub.p -k.sub.s -k.sub.i
where k=.omega.n(.omega.)/c, .DELTA.k is a measure of the phase mismatch, and n(.omega.) is the medium index of refraction for the electromagnetic energy at frequency .omega.. Phase matching is accomplished, and the efficiency of conversion is maximized, when .DELTA.k=0. This is typically achieved in nonlinear crystals by rotating the crystal to an angle at which the refractive indices for the electromagnetic energy is such that phase matching is achieved.
The phase matching requirement .DELTA.k=0 can only be met in crystals that exhibit birefringence. In isotropic media, the index of refraction, n(.omega.), is independent of the propagation direction and polarization orientation of the light wave, but is a function of the wave's frequency. In such media, the index generally increases with .omega., making phase matching impossible for real values of .omega..sub.p, .omega..sub.s, and .omega..sub.i satisfying the energy constraint. Anisotropic (birefringent) media also show the same trend in index variation with frequency, but the refractive index in such media is also a function of the propagation direction and polarization of the wave. The two polarization states of electromagnetic waves at a given frequency, .omega., in a birefringent crystal are commonly termed the ordinary and extraordinary waves and have indices n.sub.o (.omega.) and n.sub.e (.omega.), respectively. Both n.sub.o (.omega.) and n.sub.e (.omega.) are also functions of the direction of propagation with respect to the principal optical axes of the crystal, defined by the polar coordinates .theta. and .phi., and are thus fully specified as n.sub.o (w,.theta.,.phi.) and n.sub.e (w,.theta.,.phi.).
The variation of index with direction in a crystal can be spatially represented as a three-dimensional ellipsoid, which is termed the optical indicatrix. The maximum and minimum values on this indicatrix are mutually perpendicular and together with a third direction, at right angles to these two, define a Cartesian coordinate system with principal optical axes commonly referred to as x, y and z. The refractive index for any wave in any direction in the crystal, n.sub.o,e (.omega.,.theta.,.phi.), can be mathematically determined from the values of the index along these principal optical axes, n.sub.x (.omega.), n.sub.y (.omega.), and n.sub.z (.omega.).
In practice, the phase matching condition is achieved for a given parametric three-wave process by both choosing the correct polarization relation between .omega..sub.p, .omega..sub.s and .omega..sub.i and orienting the crystal to give a propagation direction with respect to the crystallographic axes which results in phase matching. The collection of propagation directions in the crystal which yield phase matching for a given set of frequencies and polarizations is termed the phase matching locus. In uniaxial crystals, this phase matching locus represents a circular section of the optical indicatrix. In biaxial crystals, the phase matching locus is more complex, but is still represented as a closed curve on the optical indicatrix. In both cases, the phase matching locus tangentially intersects at most two of the three principal optical planes.
The principal refractive indices (n.sub.x (.omega.), n.sub.y (.omega.), and n.sub.z (.omega.)) for crystals are experimentally determined and are typically only known for a few select frequencies. The values at other frequencies can be estimated by fitting the measured values to analytic expressions that approximate the relationship between wavelength and the refractive index of the crystal as a function of the wavelength corresponding to the frequencies. One of the first analytic expressions to approximate the relation between wavelength and refractive index was developed by Cauchy as a series expansion in wavelength: EQU n(.lambda.)=A+B/.lambda..sup.2 +C/.lambda..sup.4 +
If the index of refraction for a material is known at different wavelengths, the values of n(.lambda.) and .lambda. can be substituted into the Cauchy expression to yield a set of linear equations. The values for the Cauchy constants A, B, C, . . . , can then be determined by simultaneous solution of these linear equations.
The Cauchy equation shows the general trend, seen in materials at wavelengths far removed from absorption bands, of a decreasing index with increasing wavelength. Since all transmissive materials exhibit long and short wavelength absorption limits, however, the Cauchy equation is not capable of accurately representing the variation in refractive index across a full transmission "window." Near wavelengths corresponding to frequencies the material absorbs, the refractive index of the medium deviates substantially from the form predicted by the Cauchy equation. To account for the wavelength regions near absorptions, a theoretical model was developed and, in 1871, a mathematical investigation of the mechanism involved led Sellmeier to an equation that is better at characterizing the variation of refractive indices across the transmission window of materials. This equation, like Cauchy's expression, is written as a series expansion as a function of wavelength: EQU n(.lambda.).sup.2 =1+A/(1-B/.lambda..sup.2)+C/(1-D/.lambda..sup.2)+
The variables of this function are called Sellmeier coefficients. They are derived from measured values of the refractive index at different wavelengths in the same fashion as the coefficients of the Cauchy equation are obtained. The Sellmeier equation is typically only accurate in predicting principal refractive indices (and, from them, phase matching angles) in frequency regions where indices have been experimentally determined.
In KNbO.sub.3, the transmission window has a short wavelength transmission edge near 0.4 .mu.m and is transmissive over the whole visible, near and mid-infrared range out to a long wavelength transmission cut-off edge near 5 .mu.m. The published measurements of refractive indices in KNbO.sub.3, however, only cover the short wavelength range from 0.4 .mu.m to 1.064 .mu.m. Measurements of the refractive indices in the range between 1.1 .mu.m and 5 .mu.m have not been reported in the literature. The Sellmeier coefficients developed for KNbO.sub.3 are based only on the indices measured between 0.4 .mu.m and 1.064 .mu.m. Since refractive indices near the long wavelength transmission edge were not included, the accuracy of determining principal refractive indices using these coefficients diminishes at longer wavelengths.
When the Sellmeier equation is used to predict indices for wavelengths outside the region of accuracy of the Sellmeier coefficients used in the equation, the refractive indices predicted for the frequencies corresponding to those wavelengths can lead to an erroneous calculation for the correct phase matching angle for the parametric process being considered. Erroneous values of the Sellmeier coefficients can also lead to a prediction that phase matching is possible when it is not, or conversely, to a prediction that phase matching is not possible when in fact it is.
Accordingly, there is a need in the art to provide crystal orientations of KNbO.sub.3 for phase matching for frequencies outside the region where refractive indices have been experimentally determined or where phase matching has been demonstrated.